3 Answers
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Seriously, in that example image, both the star and the black hole could be moving. It just depends on which object is the dominant object in the scenario (in all cases that I can think of, that would be the black hole). It may appear to us that the black hole in the center of our galaxy is stationary, but keep in mind that our whole galaxy is moving at about 300 km/sec through the intergalactic medium, and our whole local cluster is moving at 600 km/sec (yes, that's kilometers per second!). And all the orbits of stars near a central black hole aren't always stable. There is a lot happening in those areas of space, so a star could get bumped in just the right way to go plummeting into the black hole.

Each dot represents an observation point of a star's movement through space. To say these stars are moving fast would be an understatement. Some are screaming around at 10 AU/year. It's very easy to imagine a star getting a bit too close to the black hole and getting ripped to shreds and then falling into the black hole as shown in your illustration.

No force is necessary to give a supermassive black hole, or anything else, the ability to move through space.

As Newton's Laws of Motion tell us, a body at rest tends to remain at rest, and a body in motion tends to remain in motion, unless it's acted on by some outside force. Since both the star and the black hole are in space, where there's no air and therefore no friction to slow them down, they retain whatever motion they have. The main force affecting them is gravity -- which, in this case, tends to draw them towards each other.

Objects you see in everyday life tend to come to a stop (cars stop moving when they run out of fuel, rolling stones stop rolling, etc.), but that's just because of the force of friction. In an environment like outer space, where there (usually) is no significant friction, bodies in motion can continue in motion for billions of years. Earth continues to spin, the planets continue to orbit around the Sun, and so forth -- not because anything keeps them moving, but because nothing stops them.

And there's no real distinction between rest and motion anyway. All motion is space is defined relative to some other object, or to some defined frame of reference. On Earth, we think of something as being at rest if it's not moving relative to the surface of the Earth. In space, there isn't necessary a single defined frame of reference like that. If two bodies in space are approaching each other, it's equally valid to say that one is moving and the other is at rest, or vice versa, or that they're both moving relative to an observer. That's (part of) the basis of relativity.

Any other massive object. When you are sitting on your chair, you are altering every object's track.
If the mass of an object or a cluster is a considerable number and the distance close enough, that object'd be able to move a supermassive black hole.